Download Lecture 4: Diffraction around an occulter

Hale COLLAGE (CU ASTR-7500)
“Topics in Solar Observation Techniques”
Spring 2016, Part 1 of 3: Off-limb coronagraphy & spectroscopy
Lecturer: Prof. Steven R. Cranmer
APS Dept., CU Boulder
[email protected]
http://lasp.colorado.edu/~cranmer/
Lecture 4:
Diffraction around an occulter
Brief overview
Goals of Lecture 4:
1. Understand why we need to block out the bright solar disk.
2. Derive basic physics of diffraction around an “occulter.”
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Intensity above the limb
• Visible-light emission from the
solar corona:
• K-corona due to Thomsonscattering of free electrons
• F-corona due to scattering
of inner heliospheric dust
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Intensity above the limb
• Visible-light emission from the
solar corona:
• K-corona due to Thomsonscattering of free electrons
• F-corona due to scattering
of inner heliospheric dust
• Compare with sky brightness:
• typical hazy sky
• clear day sky
(mountain-top?)
• eclipse totality sky
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
F corona = Zodiacal light
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Separating the K & F coronae
• Near the Sun, K corona is linearly polarized; F corona is unpolarized
(more about this soon…)
• K corona follows magnetic structure; F corona is ~spherically symmetric
• Original meanings: F = Fraunhofer, K = kontinuierlich (“continuous”)
• F-corona: dust is cold;
scatters full solar
spectrum, absorption
lines & all.
• K-corona: free electrons
are hot; Doppler
broadening smears out
lines in the spectrum.
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
How do we go about observing the corona?
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Off-limb geometry
• The Sun’s disk subtends solid angle Ω, angular radius = 959” = 0.267o
• Our goal: observe light coming
from larger elongation angles
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Off-limb geometry
• For observations not too far from the Sun, “observing rays” are nearly parallel:
x = Line of sight (LOS)
impact parameter in the
“plane of sky” (POS)
heliocentric radius
of any point along
the LOS
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
What can an “artificial eclipse” achieve?
i.e., why do we need to block out the Sun?
• Even if every ray is separable from every other ray,
the huge dynamic range of intensities is usually too
much for a detector to handle.
• Of course, the rays get mixed up by diffraction.
geometrical
optics
physical
optics !
Trade-offs:
• The more we occult the Sun, the better we can beat down diffraction.
• More over-occulting → we can’t see regions nearest to the Sun.
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Basic optical paths (linear cut)
• We want to block rays from the solar disk, while letting in coronal light:
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Basic optical paths (linear cut)
• In reality, some fraction of the disk light is diffracted around the occulter edge
and be reflected by the mirror, thus contaminating the coronal light.
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Diffraction
• Born & Wolf: Principles of Optics, Hecht: Optics, ASTR-5550, PHYS-4510
• Interference of electromagnetic waves when they interact with “obstacles”
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
The Huygens-Fresnel principle
• Point-sources of radiation emit spherical
wave-fronts.
Huygens:
• Every point on a spherical wave-front (surface
of constant phase) acts as a point-source for
secondary spherical waves.
Fresnel:
• E-field at any point can be constructed as the
superposition of all “incoming” wave-fronts.
Kirchhoff:
• Above is a direct consequence of the wave
equation derivable from Maxwell’s equations.
Lecture 4: Diffraction around an occulter
In vacuum, the wave-fronts
coalesce in predictable
ways…
Hale COLLAGE, Spring 2016
Diffraction: aperture geometry
• Goal: Given E(x,y,z,t) at source P0 , compute E(x,y,z,t) at observation point P.
• Note: distances r & s vary a lot for different points Q inside the aperture;
distances r′ & s′ are known & fixed.
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• A spherical wave starts at P0
• The field at point Q in the aperture (at distance r from P0) is:
• Collect time-dependence into amplitude:
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• Fresnel found that the differential amount of electric field that reaches point P
after passing through a differential area (dA) of the aperture is:
i.e.,
• But what is the normalization constant K ? Units: 1 / Length
• It tells us the relative efficiency of the secondary-wave generation.
• Two ways to derive it: 1. Forget Huygens-Fresnel, use Maxwell’s eqns!
2. Assume dA covers all space… EP = unobstructed E
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• In any case,
χ = “scattering angle” (prevents secondary waves going back to the source)
In telescopes, these angles are usually small; assume χ ≈ 0 .
• If the angles are all small, then that means the aperture diameter (call it D) is
<< the distances r, r′, s, s′
• Thus, can we can replace r & s by r′ & s′ ?
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• In any case,
χ = “scattering angle” (prevents secondary waves going back to the source)
In telescopes, these angles are usually small; assume χ ≈ 0 .
• If the angles are all small, then that means the aperture diameter (call it D) is
<< the distances r, r′, s, s′
• Thus, can we can replace r & s by r′ & s′ ?
Lecture 4: Diffraction around an occulter
In denominator: Yes
In exponent: No!
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• We need to specify distances in
terms of a known coordinate
system…
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• Remember:
(x,y) tell us the location of point P (the detector)
(ξ,η) tell us the location of point Q (“inside the aperture”)
• We’ve already assumed that D (the maximum extent of ξ and η) is << r′ & s′
• Thus, all ratios above are << 1
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Follow the waves: P0 → Q → P
• We’re getting closer to useful expressions…
• The exp argument depends on the difference between actual & reference paths:
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Simplify some things
• Assume P0 is so astronomically distant that waves at Q are plane waves.
• The reference distance s′ (from aperture to detector) is often called R
x ≈ Rθ
D
R
• Thus, if original plane wave was unobstructed, it would reach detector with:
and thus…
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fraunhofer vs. Fresnel diffraction
• Fraunhofer: for “small” apertures, quadratic terms in ξ and η can be neglected.
→ eikf integral becomes a spatial Fourier transform of the aperture
→ appropriate for slits, pinholes, and finite-sized optics in telescopes
• Fresnel: need to keep all terms.
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fresnel diffraction: rectangular aperture
• Sneaky trick: take point P along the z-axis, but move around the aperture:
y
Thus, we care only about
η2
x
η1
ξ1
for which
ξ2
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fresnel integrals
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fresnel integrals
“Cornu spiral”
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fresnel diffraction: straight-edge
• Expand 3 of the 4 sides of the rectangular aperture to infinity…
y
η2
x
η1
ξ1
ξ2
v1 < 0 : observer (at x=0) in direct sunlight
v1 > 0 : observer in the shadow of occulter
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fresnel diffraction: straight-edge
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Fresnel diffraction: straight-edge
If the calculation is
re-done with a
circular occulter or a
circular aperture…
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Far inside the shadow…
Let’s work it out for a simple idea for a solar coronagraph…
• Visible light (λ = 500 nm)
• Typical distance between occulter & primary mirror (R = 2 m)
• Try a range of θ corresponding to coronal “elongation angles”
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Far inside the shadow…
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Far inside the shadow…
What are our options?
λ = 500 nm
R=2m
• We have to move the
occulter much further
away (like the distance of
the Moon?),
• or we need to experiment
with other shapes (other
than a straight-edge),
• or we need to just deal with
the diffracted light after it
enters the telescope.
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016
Next time
• How do real coronagraphs solve (or at least minimize)
the problem of diffraction?
Lecture 4: Diffraction around an occulter
Hale COLLAGE, Spring 2016